Negative moments of $L$-functions with small shifts over function fields

TL;DR

This paper derives asymptotic formulas and bounds for negative moments of quadratic Dirichlet L-functions over function fields with small shifts, expanding understanding of their behavior in specific parameter ranges.

Contribution

It provides new asymptotic formulas and non-trivial bounds for shifted negative moments of L-functions over function fields, especially for small shifts and moments less than one.

Findings

Asymptotic formula for the $k$-th shifted negative moment when $eta o 0$ and $k<1$.

Non-trivial upper bounds for negative moments when $eta$ is very small.

Extension of previous bounds to smaller shift ranges in the function field setting.

Abstract

We consider negative moments of quadratic Dirichlet $L$--functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $\mathbb{F}_q[x]$, we obtain an asymptotic formula for the $k^{\text{th}}$ shifted negative moment of $L(1/2+\beta,\chi_D)$, in certain ranges of $\beta$ (for example, when roughly $\beta \gg \log g/g $ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\text{th}}$ shifted negative moment when $\log(1/\beta) \ll \log g$. Previously, almost sharp upper bounds were obtained in \cite{ratios} in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon}$.